Analytical treatment of projectile motion
Consider a particle projected with a velocity u of an angle with the horizontal from earth’s surface. If the earth did not attract a particle to itself, the particle would describe a straight line; on account of attraction of earth, however, the particle describes a curved path. This curve will be proved later to be always a parabola.
Let us take origin at the point of projection and x-axis and y-axis along the surface of earth and perpendicular to it respectively as shown in a figure.
By the principle of physical independence of forces, the weight of the body only has an effect on the motion in a vertical direction. It, therefore, has no effect on the velocity of the body in the horizontal direction, and horizontal velocity, therefore, remains unaltered.
Motion in x- direction:
Motion in x – direction is motion with uniform velocity.
At t = 0, x0 = 0 and ux = u cos θ
Position after time t, x = x0 + uxt
⟹ x = (u cos θ) t …………………..(1)
Velocity of any time t, vx = ux ⟹vx= u cos θ
Motion in y-direction:
Motion in y-direction is motion with uniformly acceleration
When, t = 0, y0 = 0, uy = u sin θ and ay = -g
∴ After time‘t’, vy = uy + ay t ⟹vy = u sin θ – gt
y = y0 + uyt + ayt2 ⟹y = uyt + ayt2
y = (u sinθ) t -(gt^2)/2 …………………..(2)
Also, V_y^2 = u_y^2 + 2ayy ⟹V_y^2 = u2 sin2θ – 2 gy
Time of Flight (T):
Time of flight is the time during which particle moves from O to O’ i.e., when t = T, y = 0
∴ From equation (iv)
O = u sin θ T- gT2 T = 2usinθ/g
Range of projectile (R):
Range is horizontal distance travelled in time T.
i.e., R = x (in time T)
∴ From equation (ii)
R = ucosθ.T = ucosθ 2usinθ/g
R = (u^2 sin2θ)/g
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